Given any $m$-tuple of positive integers $(w_{1},\ldots,w_{m})$ satisfying

(1.11)

$$\sum\limits_{j=1}^{m}w_{j}=b_{-}+b_{+},$$

there exist examples in Theorem 1.1 with $d_{j}-d_{j+1}=w_{j+1}$, $j=0,\dots,m-1$. Furthermore, near each singular point $t_{j}$, exactly $w_{j}$ Taub-NUT bubbles occur.